1 Introduction
Particles probing turbulent flows of viscous fluids tend to gain energy less quickly than they lose it, at all probed scales (Xu et al. Reference Xu, Pumir, Falkovich, Bodenschatz, Shats, Xia, Francois and Boffetta2014). The outcome has been related to the occurrence of flight-crash events, meaning that particles decelerate on average faster than they accelerate, and provides clear evidence, from a Lagrangian viewpoint, that classical turbulent flows are time irreversible – that is, it shows that the energy put into the system is eventually dissipated by the action of the fluid viscosity (Pumir et al. Reference Pumir, Xu, Bodenschatz and Grauer2016).
We report here our experimental investigations on the occurrence of flight-crash events in turbulent flows of superfluid
$^{4}\text{He}$
(He II), which is a quantum liquid characterized by unique properties, such as an extremely small kinematic viscosity (Barenghi, Skrbek & Sreenivasan Reference Barenghi, Skrbek and Sreenivasan2014; Mongiovì, Jou & Sciacca Reference Mongiovì, Jou and Sciacca2018). Above 1 K, as in the present study, He II can be adequately modelled by assuming that it is made of two fluids, the viscous (normal) component and the inviscid superfluid, with the density ratio between the components being strongly temperature-dependent. Additionally, turbulent flows of He II are defined by the presence of tangles of quantized vortices, which are line singularities within the fluid and can be viewed as the carriers of the flow vorticity. Indeed, it has been shown that, in the range of investigated parameters, quantum features are apparent at scales smaller than the mean distance between the vortices, regardless of the type of investigated flow, whereas, at larger scales, a classical-like behaviour is observed, within the Lagrangian framework (La Mantia et al.
Reference La Mantia, Švančara, Duda and Skrbek2016; Švančara & La Mantia Reference Švančara and La Mantia2017).
Due to the presence of quantized vortices, energy transport mechanisms in turbulent flows of He II are therefore expected to be different from those occurring in similar flows of viscous fluids – as discussed, for example, by Clark di Leoni, Mininni & Brachet (Reference Clark di Leoni, Mininni and Brachet2017). There is specifically numerical evidence that, when the normal component is absent (that is, when the fluid viscosity is null), the energy put into the system is dissipated by sound emission, following vortex reconnections and excitation of Kelvin waves. However, above 1 K, in the two-fluid regime of He II, the liquid viscosity, carried by the normal component, cannot be neglected and it should then play a role in the mechanisms of energy dissipation, although its relevance has yet to be clarified.
We consequently decided to verify this long-held expectation by experimentally looking for signatures of flight-crash events. Relatively small solid particles are suspended in the liquid and illuminated by a planar laser sheet. The flow-induced particle motions are then collected by a digital camera and processed by using the Particle Tracking Velocimetry technique (see, for example, Guo et al. Reference Guo, La Mantia, Lathrop and Van Sciver2014). We specifically calculate the velocity increments along the particle trajectories, introduced by Lévêque & Naso (Reference Lévêque and Naso2014), and the skewness of the corresponding statistical distributions, which, in homogeneous and isotropic turbulence, was found to be negative at all probed scales, indicating that particle deceleration is on average more abrupt than acceleration.
The obtained results strongly suggest that, in turbulent flows of superfluid
$^{4}\text{He}$
, flight-crash events are less apparent than in classical turbulence, at least for scales of the order of the mean distance
$\ell$
between quantized vortices, whereas, at larger scales, the flow behaviour is classical-like; that is, the skewness of the particle velocity increment distributions is neatly negative only in the latter case. The outcome therefore indicates that the action of the fluid viscosity (that is, of the fluid normal component) is mostly relevant at scales appreciably larger than
$\ell$
, whereas at smaller scales, energy transport is instead ruled by the dynamics of the quantized vortex tangle.
2 Methods
Both thermally and mechanically driven flows of He II are investigated. The latter is generated by two square grids oscillating vertically in phase (Švančara & La Mantia Reference Švančara and La Mantia2017) and the former by a flat heater placed at the bottom of a vertical channel of square cross-section, resulting in the so-called thermal counterflow of superfluid helium (Mongiovì et al.
Reference Mongiovì, Jou and Sciacca2018). Once the heater is switched on or the grids are set into motion, the particles suspended in the bath move on average in the vertical direction and their dynamics is appreciably affected by the quantized vortex tangle, especially at scales smaller than
$\ell$
(La Mantia et al.
Reference La Mantia, Švančara, Duda and Skrbek2016).
For the thermal counterflow experiments reported here we use solid deuterium hydride particles, which have sizes of a few micrometres. Their flow-induced motions are visualized in a region approximately 1 mm thick, 13 mm wide and 8 mm high, situated sufficiently away from the flow source, in the middle of our glass channel, of 25 mm sides and 10 cm high. Considering that the flow mean direction is perpendicular to our flat heater, it is not expected that significant flows occur in planes parallel to the heat source (that is, perpendicular to the field of view), especially away from the channel walls, as in the present case; see Švančara et al. (Reference Švančara, Hrubcová, Rotter and La Mantia2018) for further details on the counterflow setup.
Additionally, it is important to highlight that the present counterflow data are obtained in the turbulent state – that is, at fluid velocities appreciably larger than the turbulence onset velocity, equal to approximately
$1~\text{mm}~\text{s}^{-1}$
for our channel (La Mantia Reference La Mantia2016). In this regime the particles move on average upward, away from the heater, and their trajectories do not follow straight lines, indicating that particle motions are affected by the presence of quantized vortices; see, for example, La Mantia (Reference La Mantia2016) for a discussion on the particle behaviour in different counterflow conditions.
Deuterium particles of similar sizes are employed for the oscillating grid experiments (Švančara & La Mantia Reference Švančara and La Mantia2017) and the studied flow region has dimensions comparable to the counterflow one. It is located between the grids, as far away as possible from the solid boundaries of the experimental volume, of 50 mm sides (the fixed distance between the grids is 70 mm). It follows that, also in this case, significant flows perpendicular to the field of view (that is, parallel to the horizontal flow generator) are not expected to occur. Thermal counterflow (oscillating grid) movies are collected at 500 (400) fps. Once particle positions are obtained from the movies, the corresponding particle velocities and velocity increments are computed as in previous studies (see, for example, Švančara et al. Reference Švančara, Hrubcová, Rotter and La Mantia2018).
Following Lévêque & Naso (Reference Lévêque and Naso2014) we focus our attention here on the longitudinal velocity increments, computed along the particle trajectories. Each increment d
$v(\unicode[STIX]{x1D70F})$
is computed as the scalar product of two vectors, the Cartesian velocity increment and the corresponding position increment;
$\unicode[STIX]{x1D70F}$
indicates the time lag between the considered velocities and positions. The normalized skewness
$Sk$
of the
$\text{d}v$
statistical distribution is obtained as
$$\begin{eqnarray}\displaystyle & \displaystyle Sk(\unicode[STIX]{x1D70F})=\frac{\langle [\text{d}v(\unicode[STIX]{x1D70F})-\text{d}v_{m}(\unicode[STIX]{x1D70F})]^{3}\rangle }{\text{d}v_{sd}(\unicode[STIX]{x1D70F})^{3}}=\frac{Sk_{d}(\unicode[STIX]{x1D70F})}{\text{d}v_{sd}(\unicode[STIX]{x1D70F})^{3}}, & \displaystyle\end{eqnarray}$$
where the symbols with subscripts
$m$
and
$sd$
denote the mean and standard deviation of the velocity increment ensemble
$\langle \text{d}v(\unicode[STIX]{x1D70F})\rangle$
at time lag
$\unicode[STIX]{x1D70F}$
, respectively.
3 Results
In figure 1 we plot
$Sk$
as a function of the time ratio
$t_{R}=\unicode[STIX]{x1D70F}/\unicode[STIX]{x1D70F}_{f}$
, where
$\unicode[STIX]{x1D70F}_{f}$
indicates a relevant flow time scale, equal to the Kolmogorov time scale
$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$
for the classical data (Lévêque & Naso Reference Lévêque and Naso2014). In the case of the He II turbulent flows we follow La Mantia & Skrbek (Reference La Mantia and Skrbek2014), and set
$\unicode[STIX]{x1D70F}_{f}=\unicode[STIX]{x1D70F}_{\ell }$
, the time needed to travel a distance equal to
$\ell$
with the mean particle velocity
$V$
, obtained at the smallest time
$\unicode[STIX]{x1D70F}_{min}$
between particle positions (that is, the time between consecutive images).
Normalized skewness
$Sk$
of the longitudinal velocity increment distribution as a function of the time ratio
$t_{R}$
, see (2.1). Filled and open circles indicate thermal counterflow and oscillating grid data, respectively; relevant experimental parameters are listed in tables 1 and 2; at least
$10^{5}$
velocity increments are employed for each experimental point. The open diamonds denote the classical data, obtained at a Taylor-based Reynolds number
$R_{\unicode[STIX]{x1D706}}$
equal to
$280$
(Lévêque & Naso Reference Lévêque and Naso2014); note that, for the oscillating grid experiments,
$R_{\unicode[STIX]{x1D706}}\approx 300$
(Švančara & La Mantia Reference Švančara and La Mantia2017).
Thermal counterflow experimental conditions: temperature
$T$
, in K; applied heat flux
$q$
, in
$\text{W}~\text{m}^{-2}$
; mean particle velocity
$V$
, in
$\text{mm}~\text{s}^{-1}$
, at the smallest time
$\unicode[STIX]{x1D70F}_{min}$
between particle positions; mean distance
$\ell$
between quantized vortices, in
$\unicode[STIX]{x03BC}\text{m}$
, estimated following the procedure outlined by Švančara et al. (Reference Švančara, Hrubcová, Rotter and La Mantia2018); time
$\unicode[STIX]{x1D70F}_{\ell }$
, in ms, needed to travel a distance equal to
$\ell$
with a velocity
$V$
.
For the thermal counterflow data we estimate the mean distance between quantized vortices following the procedure outlined by Švančara et al. (Reference Švančara, Hrubcová, Rotter and La Mantia2018) – that is, from the flatness of the particle velocity distributions. The obtained
$\ell$
values are listed in table 1 together with other relevant quantities. For the oscillating grid data we employ the
$\ell$
values estimated by Švančara & La Mantia (Reference Švančara and La Mantia2017) (see also table 2).
It is apparent from figure 1 that the results obtained in He II are strikingly different from the classical one, especially at the smallest scales, of the order of
$\unicode[STIX]{x1D70F}_{\ell }$
. At larger scales, for
$t_{R}>10$
, the trend observed for the oscillating grid data appears instead to be consistent with the classical behaviour.
In order to appreciate the latter outcome we employ the dimensional value of
$Sk$
– that is, the numerator of (2.1) – and we plot in figure 2(a) the magnitude of
$Sk_{d}$
, in
$\text{mm}^{3}~\text{s}^{-3}$
, as a function of
$t_{R}$
; note that for the classical data the investigated flow has zero mean velocity and that the corresponding root mean square (r.m.s.) velocity is equal to
$145~\text{mm}~\text{s}^{-1}$
.
We clearly observe in figure 2 that our oscillating grid results and the classical data behave in a similar fashion; that is, they both follow, at large enough times, the scaling
$|Sk_{d}|\propto t_{R}$
reported by Lévêque & Naso (Reference Lévêque and Naso2014). Additionally, taking into account that a Kolmogorov-like time scale suitable for the description of He II turbulent flows should be of the same order of the scale
$\unicode[STIX]{x1D70F}_{\ell }$
employed here (as discussed, for example, by La Mantia (Reference La Mantia2017)), we can estimate the r.m.s. value of the fluid velocity for our grid experiments. If we set the latter to
$100^{1/3}~\text{mm}~\text{s}^{-1}$
, the data neatly collapse, as shown in figure 2(b), where the chosen value is approximately four times smaller than the r.m.s. value of the particle velocity in the horizontal direction, which should be less affected by the imposed vertical motion. Considering the assumptions involved in the estimate, the displayed outcome is remarkable.
(a) Absolute value of the dimensional skewness
$Sk_{d}$
, in
$\text{mm}^{3}~\text{s}^{-3}$
, as a function of
$t_{R}$
, see (2.1). (b) Absolute value of
$Sk_{d}$
, normalized by using the root mean square value
$u_{rms}$
of the fluid velocity, as a function of
$t_{R}$
;
$u_{rms}$
is equal to
$145~\text{mm}~\text{s}^{-1}$
in the classical case and is set to
$100^{1/3}~\text{mm}~\text{s}^{-1}$
for the grid experiments. Symbols as in figure 1 in both panels.
Oscillating grid experimental conditions: symbols as in table 1;
$\ell$
was estimated by Švančara & La Mantia (Reference Švančara and La Mantia2017), where further experimental details can be found. For all cases the oscillation frequency is 3 Hz and its amplitude 10 mm.
On the other hand, it is hard to deny that a striking difference between the counterflow results and the oscillating grid data is seen in figures 1 and 2. In order to highlight it we plot in figure 3 the dimensional skewness
$Sk_{d}$
as a function of
$t_{R}$
. The outcome is also apparent if, following Xu et al. (Reference Xu, Pumir, Falkovich, Bodenschatz, Shats, Xia, Francois and Boffetta2014) and Bhatnagar et al. (Reference Bhatnagar, Gupta, Mitra and Pandit2018), we calculate the particle energy increments from their velocities. Indeed, for the skewness of the energy increments distribution, the obtained behaviour is very similar to that shown in figure 3 for
$Sk_{d}$
, if it is displayed in a similar fashion.
4 Discussion
It is now time to address the physical implications of the reported results. We start by noting that the numerical data discussed by Lévêque & Naso (Reference Lévêque and Naso2014) were obtained in three-dimensional homogeneous and isotropic turbulence, while the studied He II flows have a preferential direction of motion and are investigated in a planar region, parallel to the mean flow direction, placed as far away as possible from the experimental volume vertical walls. The obtained particle trajectories therefore occur in a plane; but, as mentioned above, we do not expect that the observed particle dynamics would be significantly affected if three-dimensional tracks were considered, due to the symmetry of the imposed flow geometries. Additionally, it has been recently reported that, for classical channel flows, the statistical distributions of particle longitudinal accelerations are characterized by negative skewness values at various distances from the wall (Stelzenmuller Reference Stelzenmuller2018). It then follows that the inhomogeneity and anisotropy of the He II flows studied here are likely not sufficient to explain the reported disagreement with respect to the classical numerical results. Similarly, we may also disregard the influence of the particle inertia because numerical evidence of flight-crash events has been found, in homogeneous and isotropic turbulence, not only for tracers (Lévêque & Naso Reference Lévêque and Naso2014), but also for large buoyant bubbles (Loisy & Naso Reference Loisy and Naso2017) and inertial particles (Bhatnagar et al. Reference Bhatnagar, Gupta, Mitra and Pandit2018).
The present experimental results consequently suggest that particles probing turbulent flows of superfluid
$^{4}\text{He}$
do not tend on average to accelerate less quickly that they decelerate, if one consider scales of the order of the mean distance between quantized vortices (that is, for
$t_{R}<10$
), whereas, at larger scales, we obtain a classical-like picture. The outcome could possibly be explained by taking into account that particles can probe individual reconnections of quantized vortices solely at sufficiently small scales (La Mantia et al.
Reference La Mantia, Švančara, Duda and Skrbek2016). It then follows that, at larger scales, the particles not only sense the fluid viscosity but also probe the tangle collective behaviour, which have been both found to result in features similar to those observed in classical turbulent flows, at least within the Lagrangian framework.
To substantiate the claim we may proceed as follows. It has been reported a few times (see, for example, Zuccher et al. (Reference Zuccher, Caliari, Baggaley and Barenghi2012) and Villois, Proment & Krstulovic (Reference Villois, Proment and Krstulovic2017)), that, during reconnection, the quantized vortices approach is slower than the separation following the event. For viscous vortex reconnection the behaviour is qualitatively similar (Hussain & Duraisamy Reference Hussain and Duraisamy2011), but the exponents of the corresponding time scalings are larger than in the quantum case. This could mean that viscous reconnections are faster, more powerful events than quantum ones.
The just-mentioned time asymmetry during reconnection implies that the initial energy carried by the vortices is not solely employed to drive the vortex motions after the event, but that it is also distributed to other processes, which, in the quantum case, can be identified with sound emission and excitation of Kelvin waves, whereas, in the classical case, these processes can be mostly related to the generation of vortical structures smaller than the original ones (see also McKeown et al. Reference McKeown, Ostilla-Mónico, Pumir, Brenner and Rubinstein2018).
However, in the quantum case, the asymmetry is solely apparent at scales larger than the vortex core size – approximately
$10^{-10}~\text{m}$
(Mongiovì et al.
Reference Mongiovì, Jou and Sciacca2018) – but still significantly smaller than the typical dimension of our flow-probing particles, approximately
$10^{-6}~\text{m}$
. Indeed, it was reported, in both numerical simulations (Zuccher et al.
Reference Zuccher, Caliari, Baggaley and Barenghi2012) and experiments (Paoletti, Fisher & Lathrop Reference Paoletti, Fisher and Lathrop2010), that, at the particle scale, the asymmetry is absent because the particles cannot sense phenomena happening at scales smaller than their size. The fact that, from the present experimental results, particles do not appear to decelerate faster than they accelerate could then be related to the time symmetry of quantum vortex reconnections at sufficiently large scales, but further investigations are required to verify the interpretation.
Additionally, as noted above, the thermal counterflow behaviour is different from that observed for the oscillating grid experiments. On the basis of, for example, figure 1, it could be argued that the two trends appear to join for
$10<t_{R}<100$
, but this should solely be regarded as a possibility needing further support. For example, one could start from the consideration that, in steady-state thermal counterflow, the vortex tangle is expected to be polarized in planes perpendicular to the mean flow direction (Mongiovì et al.
Reference Mongiovì, Jou and Sciacca2018), while this should not be the case for the mechanically driven flow considered here (Švančara & La Mantia Reference Švančara and La Mantia2017).
5 Conclusions
Our experimental results clearly indicate that energy transport mechanisms in turbulent flows of superfluid
$^{4}\text{He}$
are appreciably different from those occurring in similar flows of viscous fluids. We argued that the outcome may be related to the scale-dependent interactions between our flow-probing particles and quantized vortices. It can be said that, at large enough scales, the vortex tangle collective behaviour resembles the action of the classical viscosity, whereas, at smaller scales, we found experimental evidence that, for He II turbulent flows, the energy put into the system is not dissipated as in classical turbulent flows, but by other processes, likely related to the occurrence of quantized vortex reconnections, probed by our particles.
Acknowledgements
We thank A. Naso and E. Lévêque for providing relevant data (Lévêque & Naso Reference Lévêque and Naso2014). We are grateful to P. Hrubcová, M. Rotter and L. Skrbek for valuable help. We acknowledge the support of the Czech Science Foundation (GAČR) under grant no. 19-00939S.
